Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders

Abstract

In this paper, we systematically study gauge anomalies in bosonic and fermionic weak-coupling gauge theories with gauge group G (which can be continuous or discrete). We show a very close relation between gauge anomalies and symmetry-protected trivial (SPT) orders [also known as symmetry-protected topological (SPT) orders] in one-higher dimensions. Using such an idea, we argue that, in d space-time dimensions, the gauge anomalies are described by the elements in Free[H^{d+1}(G,R/Z)]\oplus H_π^{d+1}(BG,R/Z). The well known Adler-Bell-Jackiw anomalies are classified by the free part of the group cohomology class H^{d+1}(G,R/Z) of the gauge group G (denoted as Free[H^{d+1}(G,\R/\Z)]). We refer other kinds of gauge anomalies beyond Adler-Bell-Jackiw anomalies as nonABJ gauge anomalies, which include Witten SU(2) global gauge anomaly. We introduce a notion of π-cohomology group, H_π^{d+1}(BG,R/Z), for the classifying space BG, which is an Abelian group and include Tor[H^{d+1}(G,R/Z)] and topological cohomology group H^{d+1}(BG,R/Z) as subgroups. We argue that H_π^{d+1}(BG,R/Z) classifies the bosonic nonABJ gauge anomalies, and partially classifies fermionic nonABJ anomalies. Using the same approach that shows gauge anomalies to be connected to SPT phases, we can also show that gravitational anomalies are connected to topological orders (ie patterns of long-range entanglement) in one-higher dimension.

23 pages, RevTeX4, 3 figures, 1 table

References in corpus (18)

• Classification of topological insulators and superconductors in three spatial dimensions
• Topological Field Theory of Time-Reversal Invariant Insulators
• Three dimensional topological invariants for time reversal invariant Hamiltonians and the three dimensional quantum spin Hall effect
• Topological Phases of Noncentrosymmetric Superconductors: Edge States, Majorana Fermions, and the Non-Abelian Statistics
• Physics of three dimensional bosonic topological insulators: Surface Deconfined Criticality and Quantized Magnetoelectric Effect
• Theory and classification of interacting 'integer' topological phases in two dimensions: A Chern-Simons approach
• A classification of symmetry enriched topological phases with exactly solvable models
• Integer quantum Hall effect for bosons: A physical realization
• Twisted Quantum Double Model of Topological Phases in Two--Dimension
• Z2 topological term, the global anomaly, and the two-dimensional symplectic symmetry class of Anderson localization
• Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
• Mutual Chern-Simons theory for Z_2 topological order
• Symmetry protected Spin Quantum Hall phases in 2-Dimensions
• Chiral symmetry on the edge of 2D symmetry protected topological phases
• Chiral Lattice Gauge Theories and The Strong Coupling Dynamics of a Yukawa-Higgs Model with Ginsparg-Wilson Fermions
• Chiral Lattice Gauge Theories from Warped Domain Walls and Ginsparg-Wilson Fermions
• On the decoupling of mirror fermions
• Global gauge anomalies in two-dimensional bosonic sigma models

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